### No universal group in a cardinal

by Shelah. [Sh:1029]

Forum Math, 2016

For many classes of models there are universal members in any
cardinal lambda which ``essentially satisfied GCH'', i.e. lambda
=
2^{<= lambda} . But if the class is ``complicated enough'', e.g.
the
class of linear orders, we know that if lambda is ``regular and
not
so close to satisfying GCH'' then there is no universal member.
Here
we find new sufficient conditions (which we call the olive property),not covered by earlier cases (i.e. fail the so-called SOP_4). The
advantage of those conditions is witnessed by proving that the class
of groups satisfies one of those conditions.

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